Antarctic Whale Project: Single Species
Ben Weinstein
2016-12-14 12:34:28
0.1 By Month
0.2 CCAMLR Units
0.3 Distance
0.4 Time
0.5 Velocity
0.6 Angles
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
How did the filter change the extent of tracks?
Look at the observations were defined into tracks.
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha_mu[state[i,g,t-1]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha_mu[state[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
#ag<-
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016
##Behavioral States
gamma[1] ~ dbeta(3,2) ## gamma for state 1
dev ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2] <- gamma[1] * dev
#Intercepts
alpha_mu[1] ~ dbeta(1,1)
alpha_mu[2] ~ dbeta(1,1)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)sink()
1.4 Chains
Look at the convergence of phi, just for an example
Overall relationship between phi and state, nice test of convergence.
1.4.1 Compare to priors
1.5 Parameter Summary
## parameter par mean lower upper
## 1 alpha_mu alpha_mu[1] 0.90536385 0.877351886 0.92848037
## 2 alpha_mu alpha_mu[2] 0.04673335 0.034270429 0.06224137
## 3 gamma gamma[1] 0.85421528 0.825441581 0.88482701
## 4 gamma gamma[2] 0.29897117 0.225189373 0.36825143
## 5 theta theta[1] 0.01277826 -0.009047288 0.03507419
## 6 theta theta[2] 3.15056776 2.957943727 3.37065860
2 Behavioral Prediction
Relationship between phi and state
2.1 Spatial Prediction
2.2 By individual
Overlay phi and state
2.3 Compared to CMLRR regions
2.4 Autocorrelation in behavior
2.5 Location of Behavior
3 Overlap with Krill Fishery
## Source: local data frame [17 x 5]
## Groups: Unit, Areakm2 [8]
##
## Unit Areakm2 Year Krill Density
## <fctr> <dbl> <fctr> <dbl> <dbl>
## 1 Bransfield Strait East 28316.91 1980-2009 39365.1770 1.390165e+00
## 2 Bransfield Strait East 28316.91 2009-2016 66267.4220 2.340207e+00
## 3 Bransfield Strait West 22505.23 1980-2009 39109.4540 1.737794e+00
## 4 Bransfield Strait West 22505.23 2009-2016 334129.5481 1.484675e+01
## 5 Drake Passage East 16679.78 1980-2009 200773.1174 1.203692e+01
## 6 Drake Passage East 16679.78 2009-2016 29296.7391 1.756423e+00
## 7 Drake Passage West 16067.73 1980-2009 515394.5366 3.207638e+01
## 8 Drake Passage West 16067.73 2009-2016 60794.8695 3.783663e+00
## 9 Drake Passage West 16067.73 NA 52.0590 3.239973e-03
## 10 AP East 57248.36 1980-2009 4551.0500 7.949660e-02
## 11 Elephant Island 36693.83 1980-2009 316615.9470 8.628589e+00
## 12 Elephant Island 36693.83 2009-2016 3721.3692 1.014168e-01
## 13 AP Pelagic Area 441721.50 1980-2009 60383.9770 1.367015e-01
## 14 AP Pelagic Area 441721.50 2009-2016 848.3244 1.920496e-03
## 15 AP Pelagic Area 441721.50 NA 16.7160 3.784285e-05
## 16 AP West 37218.83 1980-2009 9645.8810 2.591667e-01
## 17 AP West 37218.83 2009-2016 56013.8708 1.504987e+00
4 Time spent in grid cell
4.1 ARS
4.1.1 Time by management unit
5 Randomization tests.
5.1 Randomization I: Do whales use SSMUs non-randomly?
True time per unit
5.2 Randomization II: Do whales behave non-randomly within management units?
Given the number of observations in each unit, which are being disproportionately used for foraging?
Steps * Randomize labels of foraging and traveling at each observation * Calculate the total time foraging and traveling in each unit * Compare this null distribution to the observed value
Observed values
## Type Size PrettySize Rows Columns
## pc tbl_df 598231464 [1] "570.5 Mb" 11490000 10
## a tbl_df 157774144 [1] "150.5 Mb" 3944000 7
## data list 80981360 [1] "77.2 Mb" 10 NA
## argos array 39720952 [1] "37.9 Mb" 33 20
## obs array 39720952 [1] "37.9 Mb" 33 20
## argos_class array 19867968 [1] "18.9 Mb" 33 20
## j array 19867968 [1] "18.9 Mb" 33 20
## obs_class array 19867968 [1] "18.9 Mb" 33 20
## mdat data.frame 16339200 [1] "15.6 Mb" 49859 47
## temp ggmap 13116528 [1] "12.5 Mb" 1280 1280## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1693469 90.5 3886542 207.6 3886542 207.6
## Vcells 136618826 1042.4 298864534 2280.2 298863716 2280.2